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Learning with PurposeSlide 1
Learning with PurposeSlide 1
Chapter 6: Series-Parallel Circuits
Instructor: Jean-François MILLITHALER
http://faculty.uml.edu/JeanFrancois_Millithaler/FunElec/Spring2017
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Identifying series-parallel relationships
Most practical circuits have combinations of series and parallel components.
Components that are connected in series will share a common path.
Components that are connected in parallel will be connected across the same two nodes.
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Combination circuits
Most practical circuits have various combinations of series and parallel components. You can frequently simplify analysis by combining series and parallel components.
An important analysis method is to form an equivalent circuit.
An equivalent circuit is one that has characteristics that are electrically the same as another circuit but is generally simpler.
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Identifying series-parallel relationships
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Identifying series-parallel relationships
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Identifying series-parallel relationships
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Identifying series-parallel relationships
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Examples
RT = R1 + R5 + R4 || (R2 + R3) RT = R1 + R2 || R3 + R4 || R5
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Total resistance
Identifying the components
Total resistance RT= R1 + R2 || R3= 10 + 50 = 60 W
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Example
Find RT= 148 W
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Ohm’s law: 𝐼𝑇 =𝑉𝑆
𝑅𝑇
Find I4 = 3.45 mA (I2 = 9.29 mA)
Total Current
5 V
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Find RAB, RBC & RT
Find V1, V2, V3, V4, and V5
Voltage Relationships
V1 = V2 = 5.15 V
V3 = 4.85 V
V4 = 1.58 V
V5 = 3.27 V
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Used to precisely measure resistance
Used also in conjunction with transducers
Transducer = sensor. Change physical parameter in resistance, for example.
THE WHEATSTONE BRIDGE
Sir Charles Wheatstone, 1802-1875
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Balanced bridge when the output
VOUT = 0V
We have then V1=V2 and V3=V4
Therefore 𝑉1𝑉3
=𝑉2𝑉4
and 𝐼1𝑅1𝐼3𝑅3
=𝐼2𝑅2𝐼4𝑅4
Since 𝐼1= 𝐼3 and 𝐼2= 𝐼4
We got𝑅1𝑅3=
𝑅2𝑅4
and finally 𝑅1 = 𝑅3𝑅2
𝑅4
Bridge to find an unknown resistance
THE WHEATSTONE BRIDGE
The balanced bridge
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Determine the value of RX. The bridge is balanced (VOUT = 0 V) when RV is set at 1200 W
𝑅𝑋 = 𝑅V𝑅2𝑅4
= 1200 ∗150
100= 1800 Ω
THE WHEATSTONE BRIDGE
Example
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Unbalanced bridge when the output
VOUT ≠ 0V
Used to measure several types of physical quantities such as mechanical strain, temperature, or pressure.
Connecting the transducer in one leg of the bridge.
THE WHEATSTONE BRIDGE
The unbalanced bridge
The resistance of the transducer changes proportionally to the changes in the parameter that it is measuring.
If the bridge is balanced at a known point, then the amount of deviation indicates the amount of change in the parameter being measured.
Therefore, the value of the parameter being measured can be determined by the amount that the bridge is unbalanced
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Determine the output voltage of the temperature-measuring bridge circuit if the thermistor is exposed to a temperature of 50ºC and its resistance at 25ºC is 1.0 kΩ.
Assume the resistance of the thermistor decreases to 900 Ω at 50°C.
THE WHEATSTONE BRIDGE
Example
At 25ºC the bridge is balanced.
At 50ºC the bridge is unbalanced. We can apply the voltage-divider formula to the left and right sides.
𝑉𝐴 =𝑅3
𝑅3+𝑅𝑡ℎ𝑒𝑟𝑚𝑉𝑆 =
1𝑘Ω
1𝑘Ω+900Ω12 𝑉 = 6.32 𝑉
𝑉𝐵 =𝑅4
𝑅2+𝑅4𝑉𝑆 =
1𝑘Ω
2𝑘Ω12 𝑉 = 6. 𝑉
VOUT = VA-VB = 6.32-6 = 0.32 V at 50ºC
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What for ?
To simplify Electric Engineer’s Life !!!
Simplify a complicate series-parallel circuit into an equivalent circuit
Consists of an equivalent voltage source VTH
And an equivalent resistance RTH
THEVENIN’s Theorem
Léon Charles Thévenin, French Engineer, 1857-1926
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The Thevenin equivalent voltage VTH is the open circuit (no-load) voltage between two specified output terminals in a circuit.
The Thevenin equivalent resistance RTH is the total resistance appearing between two specified output terminals in a circuit with all sources replaced by their internal resistances (which for an ideal voltage source is zero).
THEVENIN’s Theorem
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Step 1: Find VTH = Find the voltage between A and B
Three steps: #1
Output terminalsCircuit “complicate”
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Step 2: Find RTH
Short-circuiting the battery
Find the resistance between A and B
Three steps: #2
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Step 3: Combining both VTH and RTH
Thevenin equivalent circuit
Three steps: #3
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Depends on the viewpoint
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Depends on the viewpoint
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Thevenizing a Bridge Circuit
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For a given source voltage, maximum power is transferred from a source to a load when the load resistance is equal to the internal source resistance.
Maximum power is transferred to the load when RL = RS.
Maximum Power Transfer Theorem
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Example: Determine the load power for different values of the variable load resistance [0:125] W
Solution: Using Ohm’s law and Power Formula
For RL = 0 W
𝐼 =𝑉𝑆
𝑅𝑆 + 𝑅𝐿=
10
75 + 0= 133 𝑚𝐴
𝑃𝐿 = 𝐼2𝑅𝐿 = 1332 ∗ 0 = 0𝑊
Maximum Power Transfer Theorem
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Maximum Power Transfer Theorem
For RL = 0 W : PL = 0 W
For RL = 25 W : PL = 250 mW
For RL = 50 W : PL = 320 mW
For RL = 75 W : PL = 334 mW
For RL = 100 W : PL = 326 mW
For RL = 125 W : PL = 313 mW
RL (W)0
25 50 12510075
100
200
300
400
PL (mW)Maximum for RL=RS
Note that RS = RTH if we are using the Thevenin’s Theorem
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What is happening when there are two or more voltage sources ???
How do we calculate I2 ???
Superposition Theorem
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Superposition Theorem
Problem: Find I2
Determine RT1 and IT1
Then I2(S1)
Determine RT2 and IT2
Then I2(S2)
I2 = I2(S1)+ I2(S2)
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Find I2, the current through R2
1_ short replace VS2, find RT1, then IT1 and finally I2(S1)
RT1=232 W // IT1 = 43.1 mA // I2(S1)=25.9 mA
2_ short replace VS1, find RT2, then IT2 and finally I2(S2)
RT1=399 W // IT1 = 12.5 mA // I2(S1)=3.9 mA
I2 = I1(S1) + I2(S2) =25.9 + 3.9 = 29.8 mA
Superposition Theorem
Example
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